Framing a Cubic Equation with Given the Roots :
Here we are going to see how to construct a cubic equation with given roots.
Let us consider a general cubic equation
ax3 + bx2 + cx + d = 0
α + β + γ = -b/a
α β + β γ + γα = c/a
α β γ = -d/a
Since the degree of the polynomial equation is 3, we have a ≠ 0 and hence division by a is meaningful. If a monic cubic polynomial has roots α , β , and γ , then
coefficient of x2 = − (α + β + γ)
coefficient of x = α β + β γ + γα
constant term = −α β γ
Question 1 :
If α , β and γ are the roots of the cubic equation x3 + 2x2 + 3x + 4 = 0 , form a cubic equation whose roots are
(i) 2α , 2β , 2γ
Solution :
x3 + 2x2 + 3x + 4 = 0
By comparing the given equation with the general form of cubic equation, we get
a = 1, b = 2, c = 3 and d = 4.
Coefficient of x2 = α + β + γ = 2
Coefficient of x = α β + β γ + γα = 3
Constant term = α β γ = 4
Here α = 2α, β = 2β, γ = 2γ
Coefficient of x2 = 2α + 2β + 2γ
2(α + β + γ) = 2(2) = 4
Coefficient of x = 2α (2β) + 2β(2γ) + 2γ(2α)
= 4 (αβ) + 4(βγ) + 4(γα)
= 4 (αβ + βγ + γα)
= 4(3)
= 12
Constant term = 2α (2β) (2γ)
= 8 α β γ
= 8(4)
= 32
Hence the required equation is
x3 + 4x2 + 12x + 32 = 0
(ii) 1/α, 1/β, 1/γ
Coefficient of x2 = (1/α) + (1/β) + (1/γ)
(β γ + α γ + α β)/α β γ = 3/4
Coefficient of x = (1/α) (1/β) + (1/β) (1/γ) + (1/γ)(1/α)
= (1/α β) + (1/βγ) + (1/γα)
= (γ + α + β)/α β γ
= 2/4 = 1/2
Constant term = (1/α) (1/β) (1/γ)
= 1/αβγ
= 1/4
By applying the above values in the general form of the cubic equation, we get
x3 + (3/4)x2 + (1/2)x + (1/4) = 0
4x3 + 3x2 + 2x + 1 = 0
Hence the required cubic equation is 4x3 + 3x2 + 2x + 1 = 0.
(iii) −α, -β, -γ
Coefficient of x2 = −α + (-β) + (-γ)
= - (α + β + γ)
= -2
Coefficient of x = −α (-β) + (-β)(-γ) + (-γ) (−α)
= α β + β γ + α γ
= 3
Constant term = −α (-β) (-γ)
= - αβγ
= -4
By applying the above values in the general form of the cubic equation, we get
x3 + (-2)x2 + 3x + (-4) = 0
x3 - 2x2 + 3x - 4 = 0
Hence the required cubic equation is x3 - 2x2 + 3x - 4 = 0.
After having gone through the stuff given above, we hope that the students would have understood, "Framing a Cubic Equation with Given the Roots".
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